time series analysis - introduction - escuela técnica …€¦ · · 2012-08-01time series...

Time Series Analysis

Introduction

Andres M. Alonso Carolina Garcıa-Martos

Universidad Carlos III de Madrid

Universidad Politecnica de Madrid

June – July, 2012

Alonso and Garcıa-Martos (UC3M - UPM) Time Series Analysis June – July, 2012 1 / 21

Time Series Analysis

Course description

� The course will show how to build time series models for univariate andmultivariate time series. It brings together material previously available only in theprofessional literature and presents a unified view of the advanced proceduresavailable for time series model building.

� The course begins basic concepts in univariate time series, providing anup–to–date presentation of ARIMA models, including Kalman filter, outlieranalysis, automatic methods for building ARIMA models, and signal extraction.


Basic bibliography

Time Series Analysis: Forecasting and Control, by G.E.P. Box, G.M. Jenkins,and G. Reinsel, Prentice-Hall, 1994.

Introduction to Time Series Analysis and Forecasting, by P.J. Brockwell andR.A. Davis, Springer-Verlag, 1996.

Time Series Analysis by State Space Methods, by J. Durbin and S.J.Koopman, Oxford University Press, 2001.

Forecasting in Business and Economics, by C.W.J. Granger, Academic Press,1989.

Time Series Analysis, by J. Hamilton, Princeton University Press, 1994.

A Course in Time Series Analysis, by D. Pena, G.C. Tiao and R.S. Tsay,Wiley, 2001.


Syllabus

1 Introduction to time series

2 Descriptive analysis of a time series

3 Time series and stochastic processes

4 Autoregressive, MA and ARMA processes

5 Integrated and long memory processes

6 Seasonal ARIMA processes

7 Forecasting with ARIMA models

8 Identifying possible ARIMA models

9 Estimation and selection of ARMA models

10 Model diagnosis and prediction


1. Introduction to time series

Outline:Examples of univariate time series

Examples of multivariate time series

Software for time series analysis

Recommended readings:

� Chapter 1 of Brockwell and Davis (1996).

� Chapter 1 of Pena, Tiao and Tsay (2001).


Examples of univariate time series

� A time series is the result of observing the values of a variable over time duringregular intervals (every day, every month, every year, etc.).

Example 1

The figure shows the daily trajectory, measured in nautical leagues, of ChristopherColumbus’s fleet during his first voyage to America.

0

10

20

30

40

50

60

70

5 10 15 20 25 30

Daily leagues

� Notice that the series is stable, with valuesthat oscillate around an average daily distanceof some 30 leagues.

� The graph of this series shows no clear in-creasing or decreasing trend.

Datafile columbus.xls


Example 2

The figure gives the average monthly performance of the Madrid Stock Exchangeduring the period of 1988 to 2000 as measured by the General Index. Performancehas been calculated as the relative difference between the average value of stocksat the beginning and end of each month.

-.20

-.16

-.12

-.08

-.04

.00

.04

.08

.12

.16

88 89 90 91 92 93 94 95 96 97 98 99 00

Madrid Stock General Index (Monthly performance)

� Notice that the values of the seriesseem to move around a monthly perfor-mance of 0.00722, equivalent to 8.66%(0.00722x12=.08664) annually.

� As with the previous series, this one has afixed level, with no increasing or decreasingtrends over time.

Datafile madridstockindex1988-2000.xls


Example 3

The figure shows a series which, unlike the previous two, is not clearly stable intime, and is what we call non-stationary. The series corresponds to the Spanishpopulation over 16 years of age at the end of each quarter during the period of1977 to 2000.

2.50E+07

2.60E+07

2.70E+07

2.80E+07

2.90E+07

3.00E+07

3.10E+07

3.20E+07

3.30E+07

78 80 82 84 86 88 90 92 94 96 98 00

Spanish population over 16 years of age

� Notice in the graph that the series is notstable, since its level increases over time.We say that the series has a clear, positivetrend.

� Most economic and social series are notstationary (stable) and show trends whichare more or less constant over time.

Datafile spanishover16.xls


Example 4

The figure shows the numbers of births per year in Spain from 1946 to 2000. Thisannual series is not stable, since it does not oscillate around any fixed level.

350000

400000

450000

500000

550000

600000

650000

700000

750000

50 55 60 65 70 75 80 85 90 95 00

Numbers of births in Spain

� The series does not show a growing ordecreasing linear trend in the period.

� Births in Spain increased in the 1950sand 1960s, then stabilized from 1965 to1976 and after 1977 began to decrease.This drop continues until the end of the1990s, where a slight recovery can be ob-served.

� Hence, this series shows an evolutionarytrend or one that is changing over time.

Datafile spanishbirths.xls


Example 5

The figure gives another non-stationary series: the monthly price of wheat inValladolid (Spain) from July, 1880 to December 1890.

14

16

18

20

22

24

26

28

81 82 83 84 85 86 87 88 89 90

Wheat price in Valladolid, Spain

� Again, the series shows no evolutionaround a fixed level and has a variable levelover time.

� The price of wheat rose sharply in 1882,but later dropped and in the last years expe-rienced fewer fluctuations than in the firstyears of the sample.

Datafile wheatprice.xls


Example 6

The figure shows monthly rainfall in Santiago de Compostela (Spain) over tenyears and is an example of a seasonal time series.

0

100

200

300

400

500

600

88 89 90 91 92 93 94 95 96 97

Rainfall in Santiago de Compostela, Spain

� Notice that the series oscillates arounda central value, but with some months sys-tematically having more rain than others.

� For example, the summer months haveless precipitation than winter months.

� This effect would be visible in the graphif we marked the months, but we can seethem better by plotting graphs for eachmonth.

Datafile rainfall.xls


Example 6

The figure gives the precipitation in January and July of different years.

0

100

200

300

400

500

88 89 90 91 92 93 94 95 96 97

JANUARY JULY

� Notice that in both cases the series oscil-late around a central value, and that thereare years where both are relatively high, wetyears, compared to others where both arelow, dry years.

� Nevertheless, what stands out most inthis graph is that the average values of bothseries are very different.

� This phenomenon, where the aver-age value of the time series depends onthe month being considered, is calledseasonality.


Example 7

This series combines trends and seasonality and corresponds to monthly gasolineconsumption in Spain. We see an overall growth trend in the series, although thistrend is non-linear and combines periods of growth with others of stability anddecline.

0

100

200

300

400

500

600

700

800

900

45 50 55 60 65 70 75 80 85 90 95

GASOLINE

� Moreover, the series shows marked sea-sonality, with peaks and valleys in differentmonths.

� The peaks in the graph correspond to themonths of July and August, when gasolineconsumption increases greatly due to thearrival of tourists and summer travel.

Datafile gasoline.xls


Example 8

The figure shows both seasonality and changing level once again. This seriescorresponds to the number of work related accidents per month in Spain from1979 to 1998.

20000

40000

60000

80000

100000

120000

140000

160000

80 82 84 86 88 90 92 94 96 98

Work-related accidents in Spain

� Notice that the series has varied thetrend: first decreasing then later increas-ing.

� The series also has a regular yearly pat-tern, due to seasonality.

Datafile workaccident.xls


Examples of multivariate time series

� The time series models that we are going to look at allow us to study thedynamic relationship between two or more series.

Example 9

The figure presents the vehicle registration and gasoline consumption in Spainfrom 1960 to 1999.

2

4

6

8

10

12

14

1960 1965 1970 1975 1980 1985 1990 1995

LOG(COMSUMPTION) LOG(REGISTRATION)

� Notice that the increase in regis-tration during the first years of thesample generated a clear increase ingasoline consumption.� In these series we expect a rela-tionship that goes from the numberof automobiles to gasoline consump-tion but not in the reverse way.

Datafile registrationVSconsumption.xls


� The relationships between dynamic variables can go in both directions.

Example 10

The figure shows the evolution of weekly market share of Crest and Colgatetoothpaste taken during the weeks between January 1958 and April 1963. Bothseries are non-stationary with a level that changes over time.

.0

.1

.2

.3

.4

.5

.6

1958 1959 1960 1961 1962

Market share of Colgate Market share of Crest

� In the first period we observe an increasein Crest’s market share, which is accompa-nied by a drop in Colgate’s share.

� Around observation 136 a sharp rise oc-curs in Crest’s market share but this risedoes not produce a similar decrease in thatof Colgate, which continues a downwardtrend.

Datafile crestVScolgate.xls


� Another example where the dynamic relationship goes in both directions.

Example 11

The figure shows the logarithm of the yearly estimated population of muskratsand minks in Canada from 1848 to 1911. The two series are non-stationary, witha gradual increase in the muskrat population and small level shifts in the minkpopulation.

9

10

11

12

13

14

15

50 55 60 65 70 75 80 85 90 95 00 05 10

LOG(MINKS) LOG(MUSKRATS)

� The relationship between the two seriesis one of predator - prey; since the mink eatsthe muskrat, an increase in the muskratpopulation means more food for the minks,and their population increases as well.

� But, it is to be expected that the increasein the mink population will eventually pro-duce a drop in muskrats.

Datafile minksVSmuskrats.xls


Example 12

The figure gives wheat prices in five Spanish provinces, Avila, Palencia, Segovia,Valladolid and Zamora. The series are non-stationary and seem to evolve with asimilar pattern.

12

16

20

24

28

32

81 82 83 84 85 86 87 88 89 90

Wheat price, AvilaWheat price, PalenciaWheat price, Segovia

Wheat price, ValladolidWheat price, Zamora

� We might ask ourselves if a common evo-lutionary pattern exists in all of them.

� The identification of common factors intime series is an important question andone which will be addressed at the end ofthis course.

Datafile wheatprice.xls


Summary

Time series may or may not have a stable level over time, and if they do nothave one, they can present trends which are more or less constant.

When the level of the series is stable over time we say that the time series isstationary.

Otherwise we say it is non-stationary.

A particular case of non-stationarity is when the level varies following a cycle,which is what happens with monthly temperatures during the year.

We say then that the series is seasonal and this aspect can be combined withmore or less noticeable trends in the general level.

In addition to studying the evolution of a single series, a richer time seriesmodels should consider the dynamic relationship between two or more series.

The graph of the series is always an invaluable tool for understanding itsbehavior.



� For modelling time series it is essential to have access to a statistical packagecapable of handling the necessary calculations.

� Unfortunately no single easily accessible program exists which allows us toapply all available procedure, and in the different applications we have useddifferent programs.

Statgraphics, Minitab, SPSS: provide the basic tools for identifying andestimating a univariate model for a time series, but it does not allow us todeal with outliers, nor does it permit dynamic regression or multivariateanalysis.

TSW program is based on the TRAMO/SEATS program and permits verycomplete analyses of univariate time series. It is specially designed foranalyzing many economics series. It can be downloaded for free at the Bankof Spain (Banco de Espana):http:www.bde.esserviciosoftwaretsw.htm.



The R program includes many possibilities for more sophisticated analysisand can be programmed, which makes it more flexible. R is quite similar toS-plus and can be downloaded for free at: http:www.r-project.org.

The Matlab program is very flexible programming languages, but are lesswell suited than the above programs for routine analysis of time series.There is a free version called Octave that can be downloaded at:http:www.octave.org.

There is an interface for Matlab and TRAMO/SEATS.

The Eviews program is also easy to use, includes univariate and multivariatemodels and is oriented towards econometrics.

The SCA program is the most complete of those available, but it is not aseasy to use, although it does include excellent algorithms for maximumlikelihood estimation, processing of outliers and multivariate modeling.


time series analysis - introduction - escuela técnica …€¦ · · 2012-08-01time series...

Documents